Forecasting Malaysian Gold Using. GARCH Model
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1 Applied Mahemaical Sciences, Vol. 7, 2013, no. 58, HIKARI Ld, Forecasing Malaysian Gold Using GARCH Model Pung Yean Ping 1, Nor Hamizah Miswan 2 and Maizah Hura Ahmad 3 Deparmen of Mahemaical Sciences, Faculy of Science, Universii Teknologi Malaysia, UTM Skudai, Johor, Malaysia 1 pyppung@yahoo.com, 2 miza1208@gmail.com, 3 maizah@um.my Copyrigh 2013 Pung Yean Ping e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac The purpose of he curren sudy is o forecas he prices of Kijang Emas, he official Malaysian gold bullion. Two mehods are considered, which are Box-Jenkins Auoregressive Inegraed Moving Average (ARIMA) and Generalized Auoregressive Condiional Heeroskedasiciy (GARCH). Using Akaike's informaion crierion (AIC) as he goodness of fi measure and mean absolue percenage error (MAPE) as he forecasing performance measure, he sudy concludes ha GARCH is a more appropriae model. Analysis are carried ou by using he E-views sofware. Keywords: Box-Jenkins Auoregressive Inegraed Moving Average (ARIMA), Generalized Auoregressive Condiional Heeroskedasiciy (GARCH), volailiy 1 Inroducion One goal of ime series analysis is o forecas he fuure values of he ime series daa. In he case of Kijang Emas, he official Malaysian gold bullion coin, he forecasing of is prices is useful for invesmen purposes in Malaysia. Nor Hamizah Miswan e al. [1] developed Box-Jenkins Auoregressive Inegraed Moving Average (ARIMA) model o forecas Kijang Emas prices. Kijang emas prices however are volaile wih huge price swings. Volailiy is a condiion where he condiional variance changes beween exremely high and low values. In he lieraure, when dealing wih such series, he emphasis has been given on forecasing he volailiy or he ime-varying condiional variance of he
2 2880 Pung Yean Ping e al. series. The ARCH class of models, pioneered by Engle in 1982 and generalized by Bollerslev in 1986 are popular class of economeric models for describing a series wih ime-varying condiional variance [2]. The Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) family models were developed o capure volailiy clusering or he periods of flucuaions, and predic volailiies in he fuure [3]. Seing Box-Jenkins ARIMA as he benchmark model, he curren sudy forecas Kijang Emas prices using GARCH model. By using he E-views sofware, he GARCH model is used o provide a volailiy clusering measure of he gold series. The goodness of fi and he forecasing performances of hese models are measured by Akaike's informaion crierion (AIC) and mean absolue percenage error (MAPE) respecively. 2 Mehodology The mehods ha are used in he curren sudy are Box-Jenkins ARIMA and GARCH where he former is used as a benchmark model. The daa used are Malaysian gold prices ha are non-saionary in naure. Box-Jenkins ARIMA To apply he Box-Jenkins ARIMA procedures o such ime series, he series need o be reduced o saionariy by aking a proper degree of differencing. This resuls in a model denoed by ARIMA (p,d,) where p is he auoregressive order, is he moving average order and d is he order of differences. The ARIMA(p,d,) can be wrien as d ( B )(1 B) y ( B) a where p p p ( B) 11B... pb is he auoregressive operaor of order p; ( B) 11 B... B is he moving average operaor of order; (1B) d is he d h difference; B is backward shif operaor; and a is he error erm a ime. GARCH The GARCH model on he oher hand, has he abiliy o model ime-varying condiional variances. The model uses pas variances and pas variance forecass o forecas fuure variances. The GARCH (p, ) model is
3 Forecasing Malaysian gold using GARCH model 2881 where u 2 h, ~ N(0,1) where,, for saionariy; p is he order of he GARCH erms 2, which is he las period forecas variance. is he order of he ARCH erms 2, which is he informaion abou volailiy from he previous period measured as he lag of suared residual from he mean euaion. Akaike Informaion Crierion (AIC) AIC is a echniue for selecing a model from a se of models o measure he goodness of fi of an esimaed saisical model. I is based on informaion heory and is a crierion ha seeks a model which has a good fi o he ruh bu few parameers. The model is chosen by minimizing he Kullback-Leibler disance beween he model and he ruh. AIC is compued as follows: AIC 2 ln 2k where is he maximized value of he likelihood funcion for he esimaed model; k is he number of free and independen parameers in he model. From several models of a given daa se, he bes model is he one which has he lowes AIC value. Forecas Accuracy Measure There are several measures for evaluaing forecass. For he curren sudy, he mean absolue percenage error (MAPE) will be calculaed. MAPE measures he accuracy of forecas in erms of percenage. The formula is as follows: n ˆ MAPE = y y / 100% n 1 y where is he acual value; is he forecas value; n is he number of periods. In comparing he performances beween wo models, he smaller he value of MAPE, he beer he model is. 3 Daa Analysis and Resuls Figure 1 plos he daily Kijang Emas prices recorded from18 July 2001 unil 25 Sepember 2012 ha are used in he sudy.
4 2882 Pung Yean Ping e al. Figure 1: Daily Kijang Emas Prices from 18 July 2001 unil 25 Sepember 2012 Trend is apparen from he plo, indicaing he necessiy for ransformaion and differencing o make he series saionary. Box-Cox ransformaion was firs applied, followed by aking he firs difference of he daa. Figure 2 illusraes he firs difference of he ransformed series. Figure 2: Firs Difference of Transformed Kijang Emas To idenify he model, he auocorrelaion funcion (ACF) and parial auocorrelaion funcion (PACF) of he ransformed daa are ploed in Figure 3.
5 Forecasing Malaysian gold using GARCH model 2883 Figure 3: ACF and PACF for Transformed Kijang Emas As idenified by Nor Hamizah Miswan e al. [1], he mos appropriae ARIMA model for his series is ARIMA(1,1,1) wih an AIC value of and forecasing error of For he GARCH analysis, he esing of saionariy was followed by he esing of volailiy. The rend ha affeced he non-saionariy of he series was firs removed by aking he firs difference of he series resuling in a series as ploed in Figure 4, wih he volailiy cluserings circled. Figure 4: Volailiy Clusering for he Differenced Kijang Emas
6 2884 Pung Yean Ping e al. Model idenificaion of he GARCH model is based on he ACF and PACF plos. GARCH (1, 1) was developed where he parameers of he model were esimaed by using maximum likelihood esimaion (MLE). Engle, he developer of ARCH and Bollerslev, he developer of GARCH have proven ha MLE was he bes esimaion mehod for hese models. The esimaes for GARCH (1, 1) model are = , 0 = 2.95E-06, 1 = and 1 = The values of mean euaions are small and posiive indicaing significan parameers. These saisfy he posiiviy consrain of GARCH model. The value of 0 1 1is less han bu close o uniy and This indicaes ha volailiy shocks are uie persisen. The coefficien of he lagged suared reurns is posiive and saisically significan indicaing ha srong GARCH effecs are apparen for he gold marke. Also, he coefficien of lagged condiional variance is significanly posiive and less han one indicaing ha he impac of old news on volailiy is significan. Higher value of indicaes a long memory in he 1 variance. The AIC value for his model is wih a forecasing error of Conclusion The kijang emas prices daa considered in he curren sudy can be characerized by GARCH (1, 1) model. Based on a lower SIC value, GARCH (1, 1) is more appropriae han ARIMA (1, 1, 1) in forecasing is fuure values. The lower value of MAPE for GARCH (1, 1) when compared o ha of ARIMA (1, 1, 1) showed ha GARCH (1, 1) is he more appropriae model. Acknowledgemen This sudy was suppored by Universii Teknologi Malaysia and he Minisry of Higher Educaion (MOHE), Malaysia. References [1] Nor Hamizah Miswan, Pung Yean Ping and Maizah Hura Ahmad, On Parameer Esimaion for Malaysian Gold Prices Modelling and Forecasing, Inernaional Journal of Mahemaical Analysis, 7 (22), 2013, [2] R. F. Engle, An Inroducion o he Use of ARCH/GARCH models in Applied Economerics, Journal of Business, New York (1982). [3] T. Bollerslev, Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 31, 1986, Received: March 11, 2013
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